CHAPTER 4. Results and Validation
4.4 Insulated Box containing Sphere Electrolyte Interface
Consider a rectangular box centered vertically along the z-axis with a length of 2 m and side lengths of 1 m with a fictitious surface at the center of the structure, as shown in Figure 4.14. A spherical fictitious surface is centered in the structure with a radius of 0.25 m. Electrodes are placed at the top z =2 1 m and bottom of the rectangular box z = −0 1 m , sourcing and sinking 1 A, respectively. The surfaces on the sides of the structure are insulating material. The structure was analyzed using basis orders p =0,1,2.
In Figure 4.15 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis. The electrolyte filling the boxes has a conductivity of σ1 =σ2 =4 S/m. The results match well with the analytic solution but have some error in the electric field when observing the fields closer to the fictious sphere. This may be because the field on the curved surfaces of the sphere varies more rapidly which makes the approximation more inaccurate as the field point approaches a surface of the sphere. The error decreases significantly as the basis order is increased.
In Figure 4.16 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis. In this case, the bottom box and the bottom half of the sphere are filled with an electrolyte of conductivity σ1 =2 S/m and the top box along with the top half of the sphere are filled with an electrolyte of conductivity σ2 =3 S/m. The results are in good agreement with the analytic solution in (4.1) and (4.2). The results also show the same type of error as discussed in Figure 4.15 (a) and (b) with the electric field approximation losing accuracy as it approaches the curved surfaces of the sphere at a low basis order.
32
Figure 4.14: A two-meter long brick with two homogeneous electrolytes and a spherical electrolyte interface.
33 (a)
(b)
Figure 4.15: (a) Electric potential vs. position and (b) electric field vs. position for various basis orders p and conductivity σ1 =σ2 =4 S/m and meshed using quadrilateral
cells of edge length 0.1 m.
34 (a)
(b)
Figure 4.16: (a) Electric potential vs. position and (b) electric field vs. position for various basis orders p and conductivity σ1=2 S/m, and σ2 =3 S/m and meshed using
quadrilateral cells of edge length 0.1 m.
35 4.5 Box with Linearly Varying Conductivity
Consider a cubic box aligned vertically with the z-axis and with sides of length 1 m. The bottom of the box is at z =0 m and the top of the box is at z =1 m. Electrodes are placed at the top of the box and the bottom of the box, sourcing and sinking, respectively, 1 A. The surfaces on the sides are insulating. The electrolyte has a linear conductivity profile σ
( )
r =σb+(
σ σt− b)
z where σb =1 S/m and σt =2 S/m. Theaverage potential over the bottom electrode is forced to be zero.
The analytic electric potential and electric field inside the box are ( )
( )
ln ( )respectively, where IT is the electrode current and S is the cross-sectional area of the box.
The cubic box is approximated by being divided evenly along z into N boxes where
n n
z = N for n= locates the surfaces bounding the regions in z. The conductivity 0, ,N of the nth box is σn =σ
(
rcentern)
which is the average of the actual conductivity in the nth box and where rcentern is the center of the nth box for n= . A set of 5 models, 1, ,N illustrated in Figure 4.17, was analyzed for N = . 1, ,5In Figure 4.18 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis with N = regions. The electrolyte filling the 1 box has a conductivity of σ1 =1.5 S/m. The results do not converge well to the analytic solution since a single, homogeneous electrolyte is a poor approximation of the linear conductivity profile. The electric field can only be piecewise-constant, thus it does not represent the analytic solution well.
In Figure 4.19 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, with N = regions. The electrolytes filling the boxes have 2
36
conductivities of σ1=1.25 S/m and σ2 =1.75 S/m. The results are converging to the analytic solution. The electric field can only be piecewise-constant, thus it does not represent the analytic solution well.
In Figure 4.20 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis with N = regions. The electrolytes filling the 3 boxes have conductivities of σ1=1.166 S/m, σ2 =1.5 S/m, and σ3 =1.833 S/m. The results are converging nicely, but the electric field can only be piecewise-constant, thus it does not represent the analytic solution well.
In Figure 4.21 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis with N = regions. The electrolytes filling the 4 boxes have conductivities of σ1=1.125 S/m, σ2 =1.375 S/m, σ3 =1.625 S/m, and
4 1.875 S/m
σ = . The results are converging nicely, but the electric field can only be piecewise-constant, thus it does not represent the analytic solution well.
Finally, in Figure 4.22 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis withN = regions. The 5 electrolytes filling the boxes with conductivities of σ1 =1.1 S/m, σ2 =1.3 S/m,
3 1.5 S/m
σ = , σ4 =1.7 S/m, and σ5 =1.9 S/m. The results are converging nicely, but the electric field can only be piecewise-constant, thus it does not represent the analytic solution well.
37
(a) (b)
(c) (d)
(e)
Figure 4.17: One-meter side length box models with N =1,...,5for (a) – (e), respectively.
38 (a)
(b)
Figure 4.18: (a) Electric potential vs. position and (b) electric field vs. position for N = 1 sub-boxes and various basis orders p and meshed using quadrilateral cells of edge length
0.25 m.
39 (a)
(b)
Figure 4.19: (a) Electric potential vs. position and (b) electric field vs. position for N = 2 sub-boxes and various basis orders p and meshed using quadrilateral cells of edge length
0.25 m.
40 (a)
(b)
Figure 4.20: (a) Electric potential vs. position and (b) electric field vs. position for N = 3 sub-boxes and various basis orders p and meshed using quadrilateral cells of maximum
edge length 0.25 m.
41 (a)
(b)
Figure 4.21: (a) Electric potential vs. position and (b) electric field vs. position for N = 4 sub-boxes and various basis orders p and meshed using quadrilateral cells of maximum
edge length 0.25 m.
42 (a)
(b)
Figure 4.22: (a) Electric potential vs. position and (b) electric field vs. position for N = 5 sub-boxes and various basis orders p and meshed using quadrilateral cells of maximum
edge length 0.25 m.
43 4.6 Box with Aperture
Now, consider a box with side lengths of 0.5 m and a wall thickness of 7.5 mm. A square aperture of 0.1 m exists at the center of the top surface of the box. An exterior box with side lengths of 1.5 m encompasses the smaller box. Two cylindrical electrodes aligned vertically with the z-axis with a length of 10 cm and radius of 10 cm. The structure described is shown in Figure 4.23. One electrode is placed at the center of the box z =0 m and the other is placed at z = −0.5 m, sinking and sourcing 1 A, respectively. The sides of the smaller box with aperture and the exterior box are insulating. The electrolyte filling the interior box and exterior box has a conductivity σ =4 S/m. The structure was analyzed using basis orders p =0,1,2.
For reference, the fields are calculated with and without zoning. In the zoning problem, the electrolytes are split at the top surface of the aperture. Figure 4.24 (a) and (b), respectively, provide the computed electric potential parallel to the x-axis at z = −0.1 m within the interior box without and with zoning. Figure 4.25 (a) and (b), respectively, provide the computed electric potential parallel to the x-axis at z = −0.325 moutside the interior box without and with the zoning capabilities. Comparing this to the data to with and without the zoning method applied shows the fields match up well but the convergence is slow. Now, consider the same problem but without the exterior box. It is seen in Figure 4.26 (a) and (b) that convergence is still slow for the potential within the interior box but the potential outside of the interior box exhibits much better convergence.
44
Figure 4.23: A box with aperture inside a larger insulating box.
45 (a)
(b)
Figure 4.24: Electrical potential vs. position within the interior box (a) without using zoning and (b) using zoning for various basis orders p and both boxes are meshed using
quadrilateral cells of maximum edge length 5 cm.
46
(a)
(b)
Figure 4.25: Electric potential vs. position outside the interior box (a) without using zoning and (b) using zoning for various basis orders p and both boxes are meshed using
quadrilateral cells of maximum edge length 5 cm.
47
(a)
(b)
Figure 4.26: (a) Electrical potential vs. position within the interior box and (b) electric potential vs. position outside interior box (b) for various basis orders p calculated with zoning but without the exterior box. The interior box is meshed using quadrilateral cells
of maximum edge length 5 cm.
48
4.7 Insulated cylindrical tank filled with two electrolytes.
A study of the model from [11] is conducted with a cylinderical tank centered vertically along the z-axis with a radius of 4.572 m and a height of 2.625m. A fictious surface, shown in Figure 4.27, is added at x = 0 for the zoning analysis. Two electrodes are added within the tank, both of which are placed 0.75 m below top of the tank at z =0 m and are placed at x =1.25 m and x = −1.25 m, sinking and sourcing 4.69 mA, respectively. The walls of the tank are insulating and the electrolyte within the tank has a conductivity σ =0.135 S/m. The structure was analyzed using basis orders p =0,1,2.
In Figure 4.28 (a) and (b) are plotted the x-component of the computed electric field at a depth of 2.625 m within the tank taken parallel to the x-axis at z = −1.25 m with and without zoning. In Figure 4.29 (a) and (b) shows the z-component of the computed electric field within the tank parallel to the x-axis at z = −1.25 m with and without zoning. These results agree well with the calculation presented in [11]. Note [11] presents differential potential instead of the electric field, and those results must be appropriately scaled by the probe electrode spacing.
Figure 4.27: A cylindrical tank with a false surface placed along the y-axis.
49
(a)
(b)
Figure 4.28: Ex vs. position (a) without using zoning and (b) using zoning for various basis orders p and the tank is meshed using quadrilateral cells of maximum edge length
20 cm with a depth of 2.625 m.
50 (a)
(b)
Figure 4.29: E vs. position (a) without using zoning and (b) using zoning for various z basis orders p and the tank is meshed using quadrilateral cells of maximum edge length
20 cm with a depth of 2.625 m.
51 4.8 Sphere encapsulating an electrode
Consider a problem with a pair of cylindrical electrodes are placed at x =0.4 m and 0.4 m
x = placed at z = −0.2 m. The electrodes are both cylindrical with a radius of 10 cm and a length of 10 cm. As depicted in Figure 4.30, a fictitious surface is placed around one of the electrodes with a radius of 0.1 m. The electrodes source and sink, respectively, 10 mA. The electrodes are immersed in a homogeneous electrolyte of conductivity
4 S/m.
σ = The structure was analyzed using basis orders p =0,1,2.
In Figure 4.31 (a) and (b), respectively, are plotted the computed electric potential between the electrodes parallel to the x-axis at z = −0.2 m with and without zoning, respectively. In Figure 4.32 (a) and (b), respectively, are plotted the computed electric field along the same line with and without zoning. The results match well and exhibits good convergence in both field calculations.
Figure 4.30: A pair of electrodes with a fictious sphere encompassing an electrode.
52
(a)
(b)
Figure 4.31: Electric potential vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length
0.25 m.
53 (a)
(b)
Figure 4.32: Electric field vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25
m.
54 4.9 Sphere encapsulating a Brick
Consider a problem with a rectangular box aligned vertically with the z-axis with a length of 2 m, sides of length 1 m and a fictious sphere with a radius of 1.5 m, encapsulating the structure. This is shown in Figure 4.33. Electrodes are placed at the top of the box
100 cm
z = and bottom of the box z = −100 cm, sourcing and sinking, respectively, 1 A.
A homogeneous electrolyte fills the entire region outside the box and has a conductivity of σ =4 S/m. The surfaces on the sides of the box are insulating. The structure was analyzed using basis orders p =0,1,2.
In Figure 4.34 (a) and (b), respectively, are plotted the computed electric potential along the z-axis for 100 cm< <z 300 cm with and without using zoning. In Figure 4.35 (a) and (b), respectively, are plotted the computed electric field along the same line with and without using zoning. The results match well and exhibits good convergence in both field calculations.
Figure 4.33: A brick encapsulated by a fictious sphere.
55 (a)
(b)
Figure 4.34: Electric potential vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length
0.25 m.
56 (a)
(b)
Figure 4.35: Electric field vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25
m.
57
CHAPTER 5. CONCLUSION
In this thesis, a method is presented is to accurately find electrostatic fields, potentials, and currents in zones with piecewise-homogeneous electrolytes. A surface integral equation is formulated in terms of the boundary potentials and normal currents.
The surface integral equation is discretized using an arbitrary-order, locally corrected Nyström method in order to achieve a higher degree of accuracy. The electric field calculation provides an accurate and efficient method for calculating electrostatic fields.
This method is validated by the comparison of data computed utilizing the zoning capabilities to either analytic solutions or, when applicable, data for the same problem without the zoning analysis. The method is further verified through the comparison to literature for homogeneous electrolytes.
58 REFERENCES
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60 VITA
Christopher Keith Pratt
Education
Bachelor of Science in Electrical Engineering (Cum Laude) from The University of Kentucky, August 2014 – May 2019
Bachelor of Science in Computer Science (Cum Laude) from The University of Kentucky, August 2014 – May 2019
Employment
Graduate Research Assistant, Dr. John C. Young at The University of Kentucky, August 2019 – May 2021
Lexmark Electrical Engineer Internship, May 2019 – August 2019
Undergraduate Research Assistant, Dr. Johné Parker at University of Kentucky, May 2018 – September 2019